Reinventing the Triangles: Rule of Thumb for Assessing Detectability

Abstract: Statistical significance of network clustering has been an unresolved problem since it was observed that community detection algorithms produce false positives even in random graphs. After a phase transition between undetectable and detectable cluster structures was discovered, the connection between spectra of adjacency matrices and detectability limits were shown, and both were calculated for a wide range of networks with arbitrary degree distributions and community structure. In practice the full eigenspectrum is not known, and whether a given network has any communities within detectability regime cannot be easily established. Based on the global clustering coefficient we construct a criterion telling whether in an undirected, unweighted network there is some/no detectable community structure, or if the network is in a transient regime. The method is simple and faster than methods involving bootstrapping.